In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for $\mathbf{H}(\mathbf{curl})$ problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell's equations in heterogeneous media on general domains which may have holes. Our approach has wider applicability and theoretical justification than the well-known Hiptmair-Xu auxiliary space preconditioner, with results extending to the variable coefficient case and non-convex domains at the expense of a larger coarse space.

翻译：本文针对正Maxwell型方程的协调有限元离散系统（即$\mathbf{H}(\mathbf{curl})$问题）发展并分析了其区域分解方法。众所周知，区域分解方法的收敛性高度依赖于第二层粗空间的使用效率。我们设计了自适应粗空间，该空间补充了由标量函数梯度构成的近零空间。这类新型预处理器受子空间分解思想的启发，但基于谱粗空间构建，专为一般具有空洞的异质介质域上Maxwell方程的旋度协调离散化而设计。与知名的Hiptmair-Xu辅助空间预处理器相比，我们的方法具有更广泛的适用性和理论依据，可推广至变系数情形与非凸域，但代价是需要更大的粗空间。